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I found many function generating prime numbers but all of them uses the knowledge of n-1 prime numbers that we already found or in a more inefficient way.

My question is not about existence of prime generating function but about the efficiency of such functions. Neglecting all other overheads, just the need to use all previously found primes making any algorithm to have $\Omega(n^2)$ running time.

Is there any efficient formula? If not, is there any argument supporting that you have to use all n-1 primes?

Saravanan
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    Not that anyone has discovered. – Steven F May 09 '14 at 04:51
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    There exist explicit formulas... but they involve constants that implicitly require the knowledge of the prime you are tying to calculate so cannot be used as an efficient way to calculate the $n$'th prime. See for example http://primes.utm.edu/notes/faq/p_n.html – Winther May 09 '14 at 04:54
  • no, my question is that can we create simple formula or can't? if not, is there any intuition or proof? – Saravanan May 09 '14 at 05:01

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